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A \emph{metric space} $(X,d)$ consists of a set~$X$ on which is defined a \emph{distance function} which assigns to each pair of points of $X$ a distance between them, and which satisfies the following four axioms:

\begin{itemize}
  \item
    $d(x,y) \geq 0$ for all points $x$ and $y$ of $X$;
  \item
    $(d,y) = d(y,x)$ for all points $x$ and $y$ of $X$;
  \item
    $d(x,z) \leq d(x,y) + d(y,z)$ for all points $x$, $y$ and $z$ of $X$;
  \item
    $d(x,y) = 0$ if and only if the points $x$ and $y$ coincide.
\end{itemize}

We now list the definitions of \emph{open ball}, \emph{open set} and \emph{closed set} in a metric space.

\begin{description}
  \item[open ball]
    The \emph{open ball} of radius $r$ about any point $x$ is the set of all points of the metric space whose distance from $x$ is strictly less than $r$;
  \item[open set]
    A subset of a metric space is an \emph{open set} if, given any point of the set, some open ball of sufficiently small radius about that point is contained wholly within the set;
  \item[closed set]
    A subset of a metric space is a \emph{closed set} if its complement is an open set.
\end{description}

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